Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes
Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes
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DOI：
10.1016/j.cma.2017.03.032
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发表时间：
2017-06-15
影响因子：
7.2
通讯作者：
Hierro, Alba
中科院分区：
文献类型：
--
作者：
Badia, Santiago;Bonilla, Jesus;Hierro, Alba
这项工作致力于设计的内部惩罚间断Galerkin（dG）计划，保持最大值的原则，在离散水平的稳态传输和对流扩散问题和相应的瞬态问题的隐式时间积分。将联合收割机显式时间推进与dG空间离散相结合的单调格式是很常见的，但隐式时间推进的单调格式设计却很少，迄今为止只针对一维问题。所提出的方案是基于一个分段线性dG离散补充人工扩散，线性依赖于一个冲击检测器，确定麻烦的地区。为了定义新的冲击检测器，我们引入了离散局部极值的概念。扩散算子是一个图-拉普拉斯算子，而不是更常见的拉普拉斯算子的有限元离散，这是必要的，以保持在一般网格和多维的单调性。由此产生的非线性稳定是非光滑的，非线性求解器可能无法收敛。因此，我们提出了一个平滑（二次可微）版本的非线性稳定，这使我们能够使用牛顿线搜索非线性求解器，并显着提高非线性收敛。理论数值分析表明，所提出的计划，他们满足所需的单调性。此外，所得到的算子是Lipschitz连续的，并且存在至少一个离散问题的解，即使是在非光滑的版本。我们提供了一组数值结果来支持我们的研究结果。(C)2017爱思唯尔B. V.保留所有权利。
This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings. (C) 2017 Elsevier B.V. All rights reserved.